From zero to infinity: minimum to maximum diversity of the planet by spatio-parametric Rao's quadratic entropy

Aim: The majority of work done to gather information on Earth di- versity has been carried out by in-situ data, with known issues related to epistemology (e.g., species determination and taxonomy), spatial uncertainty, logistics (time and costs), among others. An alternative way to gather information about spatial ecosystem variability is the use of satellite remote sensing. It works as a powerful tool for attaining rapid and standardized information. Several metrics used to calculate remotely sensed diversity of ecosystems are based on Shannon's In- formation Theory, namely on the dierences in relative abundance of pixel re ectances in a certain area. Additional metrics like the Rao's quadratic entropy allow the use of spectral distance beside abundance, but they are point descriptors of diversity, namely they can account only for a part of the whole diversity continuum. The aim of this paper is thus to generalize the Rao's quadratic entropy by proposing its parameterization for the rst time. Innovation: The parametric Rao's quadratic entropy, coded in R, i) allows to represent the whole continuum of potential diversity indices in one formula, and ii) starting from the Rao's quadratic entropy, al- lows to explicitly make use of distances among pixel reflectance values, together with relative abundances. Main conclusions: The proposed unifying measure is an integration between abundance- and distance-based algorithms to map the continuum of diversity given a satellite image at any spatial scale.